IMPLICIT NEURAL SPATIAL REPRESENTATIONS FOR TIME-DEPENDENT PDES

Abstract

Numerically solving partial differential equations (PDEs) often entails spatial and temporal discretizations. Traditional methods (e.g., finite difference, finite element, smoothed-particle hydrodynamics) frequently adopt explicit spatial discretizations, such as grids, meshes, and point clouds, where each degree-offreedom corresponds to a location in space. While these explicit spatial correspondences are intuitive to model and understand, these representations are not necessarily optimal for accuracy, memory-usage, or adaptivity. In this work, we explore implicit neural representation as an alternative spatial discretization, where spatial information is implicitly stored in the neural network weights. With implicit neural spatial representation, PDE-constrained time-stepping translates into updating neural network weights, which naturally integrates with commonly adopted optimization time integrators. Our approach requires neither training data nor training/testing separation. Our method is the solver itself, just like the classical PDE solver. We validate our approach on a variety of classic PDEs with examples involving large elastic deformations, turbulent fluids, and multi-scale phenomena. While slower to compute than traditional representations, our approach exhibits higher accuracy, lower memory consumption, and dynamically adaptive allocation of degrees of freedom without complex remeshing.

1. INTRODUCTION

Many science and engineering problems can be formulated as spatiotemporal partial differential equations (PDEs), F (f , ∇f , ∇ 2 f , . . . , ḟ , f , . . .) = 0, f (x, t) : Ω × T → R d . ( ) where Ω ∈ R m and T ∈ R are the spatial and temporal domains, respectively. Examples include the inviscid Navier-Stokes equations for fluid dynamics and the elastodynamics equation for solid mechanics. To numerically solve these PDEs, we oftentimes introduce temporal discretizations, {t n } T n=0 , where T is the number of temporal discretization samples and ∆t = t n+1 -t n is the time step size. The solution to Equation (1) then becomes a list of spatially dependent vector fields: {f n (x)} T n=0 . Traditional approaches represent these spatially dependent vector fields using grids, meshes, or point clouds. For example, the grid-based linear finite element method (Hughes, 2012) defines a shape function N i on each grid node and represents the spatially dependent vector field as f n (x) = P i=1 f n i N i , where P is the number of spatial samples. While widely adopted in scientific computing applications, these traditional spatial representations are not without drawbacks: 1. Spatial discretization errors abound in fluid simulations as artificial numerical diffusion (Lantz, 1971 ), dissipation (Fedkiw et al., 2001 ), and viscosity (Roache, 1998) . These errors also appear in solid simulations as inaccurate collision resolution (Müller et al., 2015) and numerical fractures (Sadeghirad et al., 2011) . 2. Memory usage spikes with the number of spatial samples P (Museth, 2013). 1

